x, a = 0, b = 1, (c) f(x

MATHEMATICAL ANALYSIS PROBLEMS LIST 12

4.01.10

(1) Find the formula for Cn = Xn

i=1

b − a n f¡

a + ib − a n

¢, and the compute lim

n→∞Cn: (a) f(x) = 1, a = 5, b = 8, (b) f(x) = x, a = 0, b = 1,

(c) f(x) = x, a = 1, b = 5, (d) f(x) = x², a = 0, b = 5, (e) f(x) = x³, a = 0, b = 1, (f) f(x) = 2x + 5, a = −3, b = 4, (g) f(x) = x²+ 1, a = −1, b = 2, (h) f(x) = x³+ x, a = 0, b = 4, (i) f(x) = e^x, a = 0, b = 1.

(2) Compute the following denite integrals by constructing a sequence of partitions of the interval, corresponding Riemann sums, and their limits:

(a) Z ₄

2

x¹⁰dx, (xi = 2 · 2^i/n), (b) Z _e

1

log(x)

x dx, (xi = e^i/n), (c)

Z ₂₀

0

x dx, (d)

Z ₁₀

1

e^2xdx, (e)

Z ₁

0

√3

x dx, (xi = _nⁱ³3), (f) Z ₁

−1

|x| dx, (g)

Z ₂

1

dx

x dx, (xi = 2^i/n), (h) Z ₄

0

√x dx, (xi = ⁴ⁱ_n2²).

(3) Compute the denite integrals:

(a) Z _π

−π

sin(x²⁰⁰⁷) dx, (b) Z ₂

0

arctan([x]) dx, (c)

Z ₂

0

[cos(x²)] dx, (d) Z ₁

0

√1 + x dx,

(e)

Z ₋₁

−2

1

(11 + 5x)³dx, (f) Z ₂

−13

1 p5

(3 − x)⁴ dx, (g)

Z ₁

0

x

(x²+ 1)² dx, (h) Z ₃

0 sgn (x³− x) dx, (i)

Z ₁

0

x e^−xdx, (j)

Z _π/2

0

x cos(x) dx, (k)

Z _e−1

0

log(x + 1) dx, (l) Z _π

0

x³ sin(x) dx, (m)

Z ₉

4

√x

√x − 1dx, (n)

Z _e³

1

1 xp

1 + log(x)dx, (o)

Z ₂

1

1

x + x³ dx, (p)

Z ₂

0

√ 1

x + 1 +p

(x + 1)³ dx, (q)

Z ₅

0

|x²− 5x + 6| dx, (r) Z ₁

0

e^x

e^x− e^−xdx,

1

(s) Z ₂

1

x log₂(x) dx, (t)

Z ^√₇

0

x³

√3

1 + x² dx, (u)

Z _6π

0

| sin(x)| dx, (w)

Z _π/2

0

cos(x) sin¹¹(x) dx, (x)

Z _{log 5}

0

e^x√ e^x− 1

e^x+ 5 dx, (y) Z _π

−π

x²⁰⁰⁷cos(x) dx, (z)

Z _2π

0

(x − π)²⁰⁰⁷cos(x) dx. (4) Prove the following estimates:

(a)

Z _π/2

0

sin(x)

x dx < 2, (b) 1

5 <

Z ₂

1

1

x²+ 1dx < 1 2, (c) 1

11 <

Z ₁₀

9

1

x + sin(x)dx < 1

8, (d) Z ₂

−1

|x|

x²+ 1dx < 3 2, (e)

Z ₁

0

x(1 − x^99+x) dx < 1

2, (f) 2√

2 <

Z ₄

2

x^1/xdx, (g) 5 <

Z ₃

1

x^xdx < 31, (h) Z ₂

1

1

xdx < 3 4. (5) Compute the following limits:

(a) lim_n→∞¡₁

n+ _n+1¹ +_n+2¹ +_n+3¹ + · · · + _2n¹ ¢ , (b) lim

n→∞

¡₁20+2²⁰+3²⁰+···+n²⁰ n²¹

¢, (c) lim

n→∞

¡ ₁

n² +_(n+1)¹ 2 +_(n+1)¹ 2 + _(n+3)¹ 2 + · · · +_(2n)¹ 2

¢· n, (d) lim_n→∞¡ ₁

√n√

2n +^√_n^√1_2n+1 + ^√_n^√1_2n+2 + ^√_n^√1_2n+3 + · · · + ^√_n^1√_3n¢ , (e) lim_n→∞¡

sin(_n¹) + sin(²_n) + sin(³_n) + · · · + sin(ⁿ_n)¢

· ¹_n, (f) lim

n→∞

¡√4n +√

4n + 1 +√

4n + 2 + · · · +√ 5n¢

· _n^√1_n, (g) lim

n→∞

¡ ₁

√3

n+ √3 ¹

n+1 + √3 ¹

n+2+ · · · + √3¹

8n

¢· √3¹

n², (h) lim

n→∞

¡√6

n·(√³ n+√³

n+1+√³

n+2+···+√³

√ 2n) n+√

n+1+√

n+2+···+√ 2n

¢, (i) lim_n→∞¡_n

n² +_n2ⁿ+1 + _n2ⁿ+4 +_n2ⁿ+9 +_n2+16ⁿ + · · · + _n2+n^{n 2}

¢, (j) lim

n→∞

¡₄

5n+ _5n+3⁴ +_5n+6⁴ + _5n+9⁴ + · · · + _26n⁴ ¢ , (k) lim

n→∞

¡ ₁

7n + _7n+2¹ +_7n+4¹ + _7n+6¹ + · · · + _9n¹ ¢ , (l) lim_n→∞¡ ₁

7n² + _7n2¹+1 +_7n2¹+2 +_7n2¹+3 + · · · + _8n¹2

¢,

(m) lim

n→∞

1 n

¡e√₁

n + e√₂

n + e√₃

n + · · · + e√_n

n¢ , (n) lim_n→∞¡ ₁

√n+^√_n+3¹ +^√_n+6¹ +^√_n+9¹ + · · · + ^√1_7n¢ ₁

√n, (o) lim_n→∞¡_n2+0

(3n)³ +_(3n+1)ⁿ²⁺¹3 +_(3n+2)ⁿ²⁺²3 + _(3n+3)ⁿ²⁺³3 + · · · + ⁿ_(4n)²⁺ⁿ3

¢, (p) lim

n→∞

¡ _n

2n² +_2(n+1)ⁿ 2 +_2(n+2)ⁿ 2 + _2(n+3)ⁿ 2 + · · · + _50nⁿ2

¢, (r) lim

n→∞

¡ _n

2n² + _n2+(n+1)^{n 2} + _n2+(n+2)^{n 2} + _n2+(n+3)^{n 2} + · · · + _50nⁿ2

¢.

2